论文标题
一元模块和半戈伦斯坦 - 训练模块
Monic modules and semi-Gorenstein-projective modules
论文作者
论文摘要
Gorenstein-progentive模块的类别$ {\ rm gp}（λ）$ tensor代数$λ= a \ otimes_kb $可以描述为单态类别$ {\ rm mon}（\ rm mon}（b，b，b，{\ rm gp}（\ rm gp}（a）（a）$ bp $ bp $ bp $ bp $ +特别是，Gorenstein-Prokentive $λ$ - 模型是Monic。在本文中，我们发现了半戈伦斯坦式$λ$ - 模块和$ a $模块之间的类似关系，即通过monic模块，即$ {\ rm mon}（b，\ ^\ perp a）= {\ rm mon}（b，a）戈伦斯坦（Gorenstein），然后$λ$是较弱的戈伦斯坦（Gorenstein），而每个半戈伦斯坦（Gorenstein）则是monic;而且，如果有限的无环颤抖的$ b = kq $带有$ q $，则$λ$在且仅当$ a $是弱的戈伦斯坦时，$λ$是弱的。但是，这种关系本身并没有回答是否存在非元人的双重半指注射$λ$模型。使用最近发现的双重半哥伦斯坦 - 标准$ $ a $模型的例子，我们通过明确构建一类双重半指注射$ t_2（a）$ - 模块，以一种参数，使它们不是一个莫尼克，并且不是monic and corsion and corsion noce torsion and tosion。对于Bimodules给出的三角形基质代数上的一元模块和半戈伦斯坦训练模块，也获得了相应的结果。
The category ${\rm gp}(Λ)$ of Gorenstein-projective modules over tensor algebra $Λ= A\otimes_kB$ can be described as the monomorphism category ${\rm mon}(B, {\rm gp}(A))$ of $B$ over ${\rm gp}(A)$. In particular, Gorenstein-projective $Λ$-modules are monic. In this paper, we find the similar relation between semi-Gorenstein-projective $Λ$-modules and $A$-modules, via monic modules, namely, ${\rm mon}(B, \ ^\perp A) = {\rm mon}(B, A)\cap \ ^\perp Λ.$ Using this, it is proved that if $A$ is weakly Gorenstein, then $Λ$ is weakly Gorenstein if and only each semi-Gorenstein-projective $Λ$-modules are monic; and that if $B = kQ$ with $Q$ a finite acyclic quiver, then $Λ$ is weakly Gorenstein if and only if $A$ is weakly Gorenstein. However, this relation itself does not answer the question whether there exist double semi-Gorenstein-projective $Λ$-modules which are not monic. Using the recent discovered examples of double semi-Gorenstein-projective $A$-modules which are not torsionless, we positively answer this question, by explicitly constructing a class of double semi-Gorenstein-projective $T_2(A)$-modules with one parameter such that they are not monic, and hence not torsionless. The corresponding results are obtained also for the monic modules and semi-Gorenstein-projective modules over the triangular matrix algebras given by bimodules.
